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Azumaya algebras of a ring with a finite automorphism group. (English) Zbl 0845.16030
Let $$G$$ be a finite group of automorphisms of a ring $$R$$, let $$R^G$$ be the subring of invariant elements of $$R$$, let $$C$$ be the center of $$R$$ and $$C^G = C \cap R^G$$, and let $$V$$ be the centralizer (i.e. commutant) of $$R^G$$ in $$R$$. The main results of this paper may be summarized as follows. If $$R$$ is a separable algebra over $$C$$ and $$R^G$$ is a central, separable algebra over $$C^G$$; then $$R$$ is the tensor product over $$C^G$$ of its subrings $$R^G$$ and $$V$$, and $$V$$ is a central, separable algebra over $$C$$. Now assume that $$R$$ is the tensor product over $$C^G$$ of its subrings $$R^G$$ and $$V$$. Then $$R$$ is a separable algebra over $$C$$ if, and only if, $$R^G \cdot C$$ and $$V$$ are central, separable algebras over $$C$$. The authors investigate some conditions which imply that $$R$$ is the tensor product of its subrings $$R^G$$ and $$V$$.
In theorem 3.4 of the paper, it seems that the hypothesis that some element of $$C$$ has trace 1 may be replaced by an assumption that some element of $$V$$ has trace 1; and this modification may be needed for the proof of theorem 3.5. Perhaps because of misprints, example 2 of the paper makes little sense.
##### MSC:
 16W20 Automorphisms and endomorphisms 16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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